Trigonometric $\vee$-systems and solutions of WDVV equations

We consider a class of trigonometric solutions of WDVV equations determined by collections of vectors with multiplicities. We show that such solutions can be restricted to special subspaces to produce new solutions of the same type. We find new solutions given by restrictions of root systems, as well as examples which are not of this form. Further, we consider a closely related notion of a trigonometric $\vee$-system and we show that their subsystems are also trigonometric $\vee$-systems. Finally, while reviewing the root system case we determine a version of (generalised) Coxeter number for the exterior square of the reflection representation of a Weyl group.


Introduction
The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations are a remarkable set of nonlinear third order partial differential equations for a single function F . They were discovered originally in two-dimensional topological field theories, and they are in the core of Frobenius manifolds theory [11], in which case prepotential F is a function on a Frobenius manifold M. A flat metric can be defined on M in terms of the third order derivatives of F which allows to reformulate WDVV equations as the associativity condition of a multiplication in a family of Frobenius algebras defined on the tangent planes T * M. Structure constants of the multiplication are also given in terms of the third order derivatives of F .
There is a remarkable class of polynomial solutions of WDVV equations, which corresponds to (finite) Coxeter groups W [11]. In this case the space M is the space of W-orbits in the reflection representation of W. Prepotential F is then a polynomial in the flat coordinates of the metric known as Saito metric.
For any Frobenius manifold there is an almost dual Frobenius manifold introduced by Dubrovin in [13]. Prepotentials for almost dual structures of the polynomial Frobenius manifolds can be expressed in a simple form where A = R is the root system of the group W. In this case the constant metric is the W-invariant form on the vector space V of the reflection representation of the group W. Such solutions F of WDVV equations appear in four-dimensional Seiberg-Witten theory as perturbative parts of Seiberg-Witten prepotentials. Thus Marshakov, Mironov and Morozov found them for classical root systems in [22,23]. Solutions (1.1) for non-classical root systems were found by Gragert and Martini in [24].
Veselov found solutions F of the form (1.1) for some not fully symmetric configurations of covectors A ⊂ V * , and he introduced the notion of a ∨-system [32] formulated in terms of linear algebra. A configuration of vectors A is a ∨-system exactly when the corresponding prepotential (1.1) satisfies WDVV equations [18,32]. This property can also be reformulated in terms of flatness of a connection on the tangent bundle T * V [33]. A closely related notion of the Dunkl system was introduced and studied in [10]. That structure for complex reflection groups was investigated further in [3] in relation with Frobenius manifolds theory.
The class of ∨-systems is closed under the natural operations of taking subsystems [18] and under restriction of a system to the intersection of some of the hyperplanes α(x) = 0, where α ∈ A [17]. The class of ∨-systems contains multi-parameter deformations of the root systems A n and B n ( [9], see also [18] for more examples). The underlying matroids were examined in [28]. The problem of classification of ∨-systems remains open.
In this work we are interested in the trigonometric solutions of WDVV equations which have the form where f ′′′ (z) = cot z, c α ∈ C, and Q = Q(x, y) is a cubic polynomial which depends on the additional variable y ∈ C. Such solutions appear in five-dimensional Seiberg-Witten theory as perturbative parts of prepotentials [23]. Solutions of the form (1.2) for (non-reduced) root systems A = R of Weyl groups and W-invariant multiplicities c α were found by Hoevenaars and Martini in [21,25]. They appear as prepotentials for the almost dual Frobenius manifold structures on the extended affine Weyl groups orbit spaces [12,14]-see [27] for type A n . In some cases such solutions may be related to the rational solutions (1.1) by twisted Legendre transformations [27].
Bryan and Gholampour found another remarkable appearance of trigonometric solutions (1.2) in geometry as they studied quantum cohomology of resolutions of A, D, E singularities [8]. The associative quantum product on these cohomologies is governed by the corresponding solutions F trig with A = A n , D n , E n respectively.
Solutions of WDVV equations of the form (1.2) without full Weyl symmetry were considered by one of the authors in [19] where the notion of a trigonometric ∨-system was introduced and its close relation with WDVV equations was established. A key difference with the rational case is the existence of a rigid structure of a series decomposition of vectors from A which generalizes the notion of strings for root systems.
Many-parameter deformations of solutions F trig for the classical root systems were obtained by Pavlov from reductions of Egorov hydrodynamic chains [26]. Closely related many-parameter family of flat connections in type A n was considered by Shen in [29,30].
Study of the trigonometric and rational cases is related since if a configuration A with collection of multiplicities c α , α ∈ A is a trigonometric ∨-system then configuration √ c α α is a rational one [19]. However, due to the presence of the extra variable y in the trigonometric case it is already nontrivial for dim V = 2 while the smallest nontrivial dimension of V in the rational case is 3. There is also an important class of elliptic solutions of WDVV equations, which was considered by Strachan in [31] where, in particular, certain solutions related to A n and B n root systems were found. The prepotentials appear as almost dual prepotential associated to Frobenius manifold structures on A n and B n Jacobi groups orbit spaces [4,5]. Such solutions appear also in six-dimensional Seiberg-Witten theory [7].
In this paper we study trigonometric solutions F trig of the form (1.2) of WDVV equations. In section 2 we recall the notion of a trigonometric ∨-system and revisit its close relation with solutions of WDVV equations.
We investigate operations of taking subsystems and restrictions in section 3, 4. We show that a subsystem of a trigonometric ∨-system is also a trigonometric ∨-system, and that one can restrict solutions of WDVV equations of the form (1.2) to the intersections of hyperplanes to get new solutions.
In section 5 we find solutions F trig for the root system BC n which depend on three parameters. By applying restrictions we obtain in sections 5 and 6 multi-parameter families of solutions F trig for the classical root systems thus recovering and extending results from [26]. In the case of BC n we get a family of solutions depending on n + 3 parameters which can be specialized to Pavlov's (n+1)-parametric family from [26]. A related multi-parameter deformation of BC n solutions (1.2) when Q depends on x variables only was obtained in [1] by similar methods.
In section 7 we consider solutions F trig for n ≤ 4. We show that solutions with up to five vectors on the plane belong to deformations of classical root systems. We also get new examples of solutions F trig of the form (1.2) some of which cannot be obtained as restrictions of solutions (1.2) for the root systems.
In section 8 we revisit solutions F trig for the root systems studied in [8,21,25,29,30]. The polynomial Q in this case depends on a scalar γ (R,c) which is determined in these papers for any invariant multiplicity function c : R → C. We give a formula for γ (R,c) in terms of the highest root of R generalizing a statement from [8] for special multiplicities. We also find a related scalar λ (R,c) which is invariant under linear transformations applied to the root system R. This scalar may be thought of as a version of generalized Coxeter number (see e.g. [20]) for the irreducible W-module Λ 2 V since it is given as a ratio of two canonical W-invariant symmetric bilinear forms on Λ 2 V .

Trigonometric ∨-systems and WDVV equations
Let V be a vector space of dimension N over C and let V * be its dual space. Let A be a finite collection of covectors α ∈ V * which belongs to a lattice of rank N.
Let us also consider a multiplicity function c : A → C. We denote c(α) as c α . We will assume throughout that the corresponding symmetric bilinear form is non-degenerate. We will also write G A for G (A,c) to simplify notations. The form G A establishes an isomorphism φ : V → V * , and we denote the inverse Let U ∼ = C be a one-dimensional vector space. We choose a basis in V ⊕ U such that e 1 , . . . , e N is a basis in V and e N +1 is the basis vector in U, and let x 1 , . . . , x N +1 be the corresponding coordinates. We represent vectors x ∈ V, y ∈ U as x = (x 1 , ..., x N ) and where λ ∈ C * and function f (z) = 1 6 iz 3 + 1 4 Li 3 (e −2iz ) satisfies f ′′′ (z) = cot z. The WDVV equations is the following system of partial differential equations where F i is (N + 1) × (N + 1) matrix with entries (F i ) pq = ∂ 3 F ∂x i ∂xp∂xq (p, q = 1, . . . , N + 1). Let e 1 , ..., e N be the basis in V * dual to the basis e 1 , ..., e N ∈ V. Then for any covector where we denoted by α both column and row vectors α = (α 1 , ..., α N ), and α ⊗ α is N × N matrix with matrix entries (α ⊗ α) jk = α j α k . Let us define where i, j = 1, . . . , N + 1. Now we will establish a few lemmas which will be useful later. The next statement is standard.
where η kl is defined in (2.4) and the summation over repeated indices here and below is assumed. It is clear from the definition that the multiplication * is commutative and distributive. The proof of the next statement is standard (see [11] for a similar statement).
Lemma 2.2. The associativity of multiplication * is equivalent to the WDVV equation (2.2).
Let us introduce vector field E by

Proposition 2.3.
Vector field E is the identity for the multiplication (2.5).
Then the product (2.5) has the following explicit form Proof. Note that η m,N +1 = 1 2 δ N +1 m for any m = 1, ..., N +1, where δ j i is the Kronecker symbol. Therefore from (2.5) we have Then we have N k,l=1 by Lemma 2.1. Also by formula (2.3) we have that The statement follows from formulas (2.7) and (2.8).
If we identify vector space V ⊕ U with the tangent space T (x,y) (V ⊕ U) ∼ = V ⊕ U, then multiplication (2.6) can also be written as Now for each vector α ∈ A let us introduce the set of its collinear vectors from A: Let δ ⊂ δ α and α 0 ∈ δ α . Then for any γ ∈ δ we have γ = k γ α 0 for some k γ ∈ R. Note that k γ depends on the choice of α 0 and different choices of α 0 give rescaled collections of these parameters. Define C α 0 δ := γ∈δ c γ k 2 γ . Note that C α 0 δ is non-zero if and only if C α 0 δ = 0 for any The following proposition holds.
Proof. For any a = (a 1 , ..., a N ) The WDVV equations (2.2) are equivalent to the commutativity [F ∨ a , F ∨ b ] = 0 for any a, b ∈ V. The commutativity [F ∨ a , F ∨ b ] = 0 is then equivalent to the identities [19] γ,β∈A Let us consider terms in the left-hand side of the relation (2.11), where β or γ is proportional to α. The sum of these terms has to be regular at α(x) = 0. This implies that the product is regular at α(x) = 0. The first factor in the product (2.12) has the first order pole at α(x) = 0 by the assumption that C α 0 δα = 0 for any α ∈ A, α 0 ∈ δ α . This implies the statement.
Similarly to Proposition 2.5 the following proposition can also be established.
The proof is similar to the proof of Proposition 2.5. Indeed, we have that expression (2.12) is regular at α(x) = πm, m ∈ Z. Assumptions imply that the first factor in (2.12) has the first order pole, which implies the statement.
The WDVV equations for a function F can be reformulated using geometry of the configuration A. Such a geometric structure is embedded in the notion of a trigonometric ∨-system. Before defining trigonometric ∨-system precisely we need a notion of series (or strings) of vectors (see [19]).
For any α ∈ A let us distribute all the covectors in A \ δ α into a disjoint union of α-series where k ∈ N depends on α. These series Γ s α are determined by the property that for any s = 1, . . . , k and for any two covectors γ 1 , γ 2 ∈ Γ s α one has either γ 1 +γ 2 = mα or γ 1 −γ 2 = mα for some m ∈ Z. We assume that the series are maximal, that is if γ ∈ Γ s α for some s ∈ N, then Γ s α must contain all the covectors of the form ±γ + mα ∈ A with m ∈ Z. Note that if for some β ∈ A there is no γ ∈ A such that β ± γ = mα for m ∈ Z, then β itself forms a single α-series.
By replacing some vectors from A with their opposite ones and keeping the multiplicity unchanged one can get a new configuration whose vectors belong to a half-space. We will denote such a system by A + . If this system contains repeated vectors α with multiplicities c i α then we replace them with the single vector α with multiplicity c α := i c i α .
Definition 2.7. [19] The pair (A, c) is called a trigonometric ∨-system if for all α ∈ A and for any α-series Γ s α , one has the relation Note that if β 1 , β 2 ∈ Γ s α for some α, s, then α ∧ β 1 = ±α ∧ β 2 so the identity (2.13) may be simplified by cancelling wedge products. We also note that if A is a trigonometric ∨-system then A + is the one as well.
The close relation between the notion of a trigonometric ∨-system and solutions of WDVV equations is explained by the next theorem. Before we formulate it let us introduce two symmetric bilinear forms G Let us consider the bilinear form G (2.14) where z, w ∈ Λ 2 V. It is easy to see that for , which is a natural extension of the bilinear form G A to the space Λ 2 V. It is also easy to see that this form G (1) A is non-degenerate and that it is W-invariant. Let us also define the following bilinear form G The following statement shows that the bilinear form G A + is independent of the choice of the positive system A + .
Proof. Suppose firstly that two positive systems A (1) + for a trigonometric ∨-system (A, c) satisfy the condition A (2.16). The statement follows in this case.
In general, the system A (2) + can be obtained from the system A (1) + by a sequence of steps where in each one we replace the subset of vectors δ α with vectors −δ α and the resulting system is still a positive one. In order to see this one moves continuously the hyperplane defining A (1) + into the hyperplane A (2) + so that at each moment the hyperplane contains at most one vector from A up to proportionality. The statement follows from the case considered above.
As a consequence of Lemma 2.8 we can and will denote the form G A close relation between trigonometric ∨-systems and solutions of WDVV equations is given by the following theorem. Theorem 2.9. (cf. [19]) Suppose that a configuration (A, c) satisfies the condition C α 0 δ = 0 for all α ∈ A, δ ⊂ δ α , α 0 ∈ δ α . Then WDVV equations (2.2) for the function (2.1) imply the following two conditions: A .

Conversely, if a configuration (A, c) satisfies conditions (1) and (2) then WDVV equations (2.2) hold.
The key part of the proof is to derive trigonometric conditions from WDVV equations, which goes along the following lines (see [19] for details). By Proposition 2.6 identity (2.10) holds if tan α(x) = 0. The identity (2.10) is a linear combination of cot β(x)| tan α(x)=0 , which can vanish only if it vanishes for each α-series. Hence identity (2.10) implies relations (2.13) so A is a trigonometric ∨-system. Remark 2.10. A version of Theorem 2.9 is given in [19, Theorem 1] without specifying conditions C α 0 δ = 0. However these assumptions seem needed in general in order to derive trigonometric ∨-conditions for α-series in the case when δ α \ {±α} = ∅ as above arguments and proofs of Propositions 2.5, 2.6 explain.
An important class of solutions of WDVV equations is given by (crystallographic) root systems A = R of Weyl groups W. Recall that a root system R satisfies the property for any α, β ∈ R, and one has 2 α,β α,α ∈ Z, where ·, · is a W-invariant scalar product on V * ∼ = V. The corresponding Weyl group is generated by reflections s α , α ∈ R.
The following statement was established in [25] for the non-reduced root systems.
Theorem 2.11. (cf. [25]) Let A = R be an irreducible root system with the Weyl group W and suppose that the multiplicity function c : Let us explain a proof of this statement different from [25] by making use the notion of a trigonometric ∨-system and Theorem 2.9.
Proposition 2.12. Root system A = R with W-invariant multiplicity function c is a trigonometric ∨-system.
Proof. Fix α ∈ R. Take any β ∈ R, and let γ = s α β. Then from (2.17) we have that Hence β, γ ∈ Γ s α for some s. The bilinear form G R is W-invariant so is proportional to ·, · . Therefore we have It is easy to see that the bilinear form G (1) R is W-invariant, and the same is true for the bilinear form G R have to be proportional. By Theorem 2.9 this implies Theorem 2.12 provided that the form G (2) R is non-zero. The latter fact is claimed in [25] where the corresponding solution of WDVV equations was explicitly stated for the constant multiplicity function. It was found for any multiplicity function for the non-reduced root systems in [29,30].
It follows that a positive half A = R + of a root system R also defines a solution of WDVV equations (2.2). We find the corresponding form G R + for the root system R = BC N explicitly in section 5. We also specify corresponding constants λ = λ (R,c) for (the positive halves of) reduced root systems R in section 8. Note that λ is invariant under the linear transformations applied to A. In the root system case the scalar λ (R,c) may be thought of as a version of the (generalized) Coxeter number for the case of the representation Λ 2 V , as the usual (generalized) Coxeter number can also be given as a ratio of two W-invariant forms on V ( [6,20]).

subsystems of trigonometric ∨-systems
In this section we consider subsystems of trigonometric ∨-systems and show that they are also trigonometric ∨-systems. An analogous statement for the rational case was shown in [18] (see also [16] is a disjoint union of two non-empty subsystems, and it is called irreducible otherwise. Consider the following bilinear form on V associated with a subsystem B: Let us prove some lemmas which will be useful for the proof of the main theorem of this section. (2) For any α ∈ B, α ∨ is an eigenvector for M.
(3) The space W ∨ can be decomposed as a direct sum where λ i ∈ C are distinct, and the restriction M| U λ i = λ i I, where I is the identity operator.
Proof. Let u, v ∈ V. We have which proves the first statement. Let us consider a two-dimensional plane π ⊂ V * such that π contains α and another covector from B which is not collinear with α. Let us sum up ∨-conditions (2.13) over αseries which belong to the plane π. We get that for some λ π ∈ C. Let us now sum up relation (3.3) over all such two-dimensional planes π which contain α and another non-collinear covector from B. It follows that M(α ∨ ) = λα ∨ , for some λ ∈ C, hence property (2) holds. The set of vectors {α ∨ : α ∈ B} spans W ∨ since B spans W . As α ∨ is an eigenvector for M| W ∨ for any α ∈ B we get that M| W ∨ is diagonalizable, and W ∨ has the eigenspace decomposition as stated in (3.2).
. Hence we have the required relation (3.4). Note that λ i = 0 for all i as otherwise G B | U λ i ×V = 0 which contradicts the non-isotropicity of B.
The following lemma relates vectors β ∨ B and β ∨ .
Lemma 3.4. Let A and B be as stated in Lemma 3.1. Let α ∈ B and let i ∈ N be such that α is a linear combination of β and α, we get that γ ∈ U λ i as required.
The opposite inclusion is obvious. Proposition 3.6. In the assumptions and notations of Lemma Now we present the main theorem of this section.
Theorem 3.7. Any non-isotropic subsystem of a trigonometric ∨-system is also a trigonometric ∨-system.
Proof. Let A be a trigonometric ∨-system and let B be its non-isotropic subsystem. Let α ∈ B. Then α ∨ ∈ U λ i in the decomposition (3.1) for some i. Consider an α-series Γ B α in B. Let β ∈ Γ B α . Then by Lemma 3.4 we have the following two cases.

restriction of trigonometric solutions of WDVV equations
In this section we consider the restriction operation for the trigonometric solutions of WDVV equations and show that this gives new solutions of WDVV equations. An analogous statement in the rational case was established in [17]. Let Let us denote the restriction α| W B of a covector α ∈ V * as π B (α), then Consider a point x 0 ∈ M B and tangent vectors u 0 , v 0 ∈ T x 0 M B . We extend vectors u 0 and v 0 to two local analytic vector fields u(x), v(x) in the neighbourhood U of x 0 that are tangent to the subspace W B at any point x ∈ W B ∩ U such that u 0 = u(x 0 ) and v 0 = v(x 0 ). Consider the multiplication * given by (2.9). We want to study the limit of u(x) * v(x) when x tends to x 0 . The limit may have singularities at x ∈ W B as cot α(x) with α ∈ B is not defined for such x. Also we note that outside W B we have a well-defined multiplication u(x) * v(x).
The proof of the next lemma is similar to the proof of [17,Lemma 1] in the rational case (see also [1]).
In particular, the product u 0 * v 0 is determined by vectors u 0 and v 0 only. Now for the subsystem B ⊂ A given by (4.1) let where k = dim W, be a basis of W. The following lemma shows that multiplication (4.3) is closed on the tangent space T * (M B ⊕ U).
Proof. Suppose that the subspace W B given by (4.2) has codimension 1 in V, and let α ∈ S.
for any z, w ∈ V. By taking z, w / ∈ Π α we derive from (4.5) that which implies the statement by Lemma 4.1.
Let us now consider W B of codimension 2. Let S = {α 1 , α 2 }. By the above arguments This proves the statement for the case when W B has codimension 2. General B is dealt with similarly.
Let us assume that G A | W B is non-degenerate. Then we have the orthogonal decomposition where α ∨ ∈ W B and w ∈ W ⊥ B . By Lemmas 4.1, 4.2 we have associative product Proof. From decomposition (4.6) we have Let us choose a basis in the space W B ⊕U such that f 1 , . . . , f n is a basis in W B , n = dim W B , and f n+1 is the basis vector in U, and let ξ 1 , . . . , ξ n+1 be the corresponding coordinates. We represent vectors ξ ∈ W B , y ∈ U as ξ = (ξ 1 , ..., ξ n ) and y = ξ n+1 . The WDVV equations for a function F : W B ⊕ U → C is the following system of partial differential equations: where F i is (n + 1) × (n + 1) matrix with entries (F i ) pq = ∂ 3 F ∂ξ i ∂ξp∂ξq (p, q = 1, . . . , n + 1). The previous considerations lead to the following theorem.
Theorem 4.4. Let B ⊂ A be a subsystem, and let S be as defined in (4.4). Assume that prepotential (2.1) satisfies WDVV equations (2.2). Suppose that C α 0 δα = 0 for any α ∈ S, α 0 ∈ δ α . Then the prepotential where α = π B (α), satisfies the WDVV equations (4.7). The corresponding associative multiplication has the form In general a restriction of a root system is not a root system, so we get new solutions of WDVV equations by applying Theorem 4.4 in this case. In sections 5, 6 and 8 we consider such solutions in more details.

BC N type configurations
In this section we discuss a family of configurations of BC N type and show that it gives trigonometric solutions of the WDVV equations. Let the set A = BC + N consist of the following covectors: Let us define the multiplicity function c : BC + N → C by c(e i ) = r, c(2e i ) = s, c(e i ± e j ) = q, where r, s, q ∈ C. We will denote the configuration (BC + N , c) as BC + N (r, s, q). It is easy to check that is assumed to be non-zero, and (e i ∧ e j ) 2 + (e i ∧ e k ) 2 + (e i ∧ e l ) 2 + (e j ∧ e k ) 2 + (e j ∧ e l ) 2 + (e k ∧ e l ) 2 We also have A corresponding to the bilinear forms G A (·, ·), G A (·, ·) respectively have the following forms: and G (2)
Proof. Let us first prove identity (5.14). Note that G A is a quadratic polynomial in r, s and q. The terms containing r 2 add up to Let us now prove identity (5.15). Note that hG (2) A is a quadratic polynomial in r, s and q and that terms containing r 2 , rs and s 2 all vanish. Terms containing rq in hG (2) A are given by  The previous proposition allows us to prove the following theorem. where h is given by (5.2), provided that q(r + 8s + 2(N − 2)q) = 0.
Proof. Firstly, BC + N (r, s, q) is a trigonometric ∨-system by Proposition 2.12. Secondly, by Proposition 5.2 we have that G A = 0 if λ is given by (5.27). The statement follows by Theorem 2.9. Theorem 5.3 gives a generalization of the results in [21], [25], [8] and [30], where, in particular, solutions of the WDVV equations for the root systems D N , B N and C N were obtained. Following [21], [25] consider the function F of N + 1 variables (x 1 , . . . , x N , y) of the form where R + is a positive half of the root system R, multiplicities c α are invariant under the Weyl group, γ ∈ C and function f given by where we redenoted variables (x, y) by ( x, y). By changing variables x = −ix, y = γλ 2h y and dividing by −λ solution (5.31) takes the form (5.28) provided that γ 2 λ 2 = −4h 3 which is satisfied for γ given by (5.30).
Let n ∈ N and let m = (m 1 , . . . , m n ) ∈ N n be such that Let us consider the subsystem B ⊂ A = BC + N given by Let us also consider the corresponding subspace W B = {x ∈ V : β(x) = 0, ∀β ∈ B}. It can be given explicitly by the equations . . , ξ n are coordinates on W B . Let us now restrict the configuration BC + N (r, s, q) to the subspace W B . That is we consider non-zero restricted covectors α = π B (α), α ∈ BC + N with multiplicities c α , and we add up multiplicities if the same covector on W B is obtained a few times. Let us denote the resulting configuration as BC n (q, r, s; m). It is easy to see that it consists of covectors where f 1 , . . . , f n is the basis in W * B corresponding to coordinates ξ 1 , . . . , ξ n . As a corollary of Theorem 4.4 and Theorem 5.3 we get the following result on (n + 3)parametric family of solutions of WDVV equations, which can be specialized to (n + 1)parametric family of solutions from [26].
Proof. We only have to check that cubic terms in (5.33) have the required form. For any ξ ∈ W B we have by formula (5.32). Hence (5.34) becomes as required.

A N type configurations
In this section we discuss a family of configurations of type A N and show that it gives trigonometric solutions of the WDVV equations. Let be the positive half of the root system A N given by Let t = c(e i − e j ) ∈ C be the constant multiplicity. The following lemma gives the relation between covectors in A and their dual vectors in V .
Proof. Let x = (x 1 , . . . , x N +1 ), y = (y 1 , . . . , y N +1 ) ∈ V. Then the bilinear form G A takes the form which implies the statement. Now we can find the forms G A corresponding to the bilinear forms G A (·, ·) respectively have the following forms: Proof. For the first equality we have For equality (6.1) we have by Lemma 6.1 that (e i ∧ e j − e i ∧ e k + e j ∧ e k ) 2 = 3(N + 1) (e i ∧ e j − e i ∧ e k + e j ∧ e k ) 2 . (6.4) Equality (6.1) follows from formulas (6.2)-(6.4).
This leads us to the following result which can also be extracted from [25].
Proof. Firstly, A is a trigonometric ∨-system by Proposition 2.12. Secondly, by Proposition 6.2 we have that which is equal to 0 for λ given by (6.5). It follows by Theorem 2.9 that F satisfies WDVV equations (2.2) as a function on the hyperplane V ⊂ C N +1 which also depends on the auxiliary variable y. Now we change variables to (x 1 , . . . , x N +1 ) by putting y = N +1 i=1 x i , which implies the statement.
Let us now apply the restriction operation to the root system A N . Let n ∈ N and m = (m 1 , . . . , m n+1 ) ∈ N n+1 be such that n+1 i=1 m i = N + 1. Let us consider the subsystem B ⊂ A given as follows: The corresponding subspace W B defined by (4.2) can be given explicitly by the equations The restriction of the configuration A + N to the subspace W B consists of the following covectors: The following result holds, which is closely related to a multi-parameter family of solutions found in [26] (see also [30]).
Proof. Let us suppose firstly that m i ∈ N for all i = 1, . . . , n + 1. Define N = −1 + n+1 i=1 m i . By Theorem 6.3 function (2.1) with A = A + N and λ given by (6.5) is a solution of WDVV equations (2.2). By Theorem 4.4 the prepotential given by as a function on W B ⊕ C satisfies WDVV equations. Note that Note also that and that By making use (6.9)-(6.11) the function (6.8) takes the form By setting y = n+1 i=1 m i ξ i and moving to variables (ξ 1 , . . . ξ n+1 ) ∈ C n+1 solution (6.12) takes the required form (6.7). The case of complex m i follows from the above since F depends on m i polynomially.
Remark 6.5. We note that Theorem 6.3 and the solution F given by (2.1) is valid if one takes any generic linear combination of coordinates x i to form the extra variable y = N +1 i=1 a i x i , a i ∈ C. The corresponding solution after restriction is given by the formula where y is a linear combination of ξ 1 , . . . , ξ n+1 , ξ i ∈ C.

further examples in small dimensions
In section 4 we presented the method of obtaining new solutions of WDVV equations through restrictions of known solutions. We applied it to classical families of root systems in sections 5, 6. Similarly, starting from any root system and the corresponding solution of WDVV equations one can obtain further solutions by restrictions. In the next proposition we deal with a family of configurations in 4-dimensional space which in general is not a restriction of a root system. Proposition 7.1. Let a configuration A ⊂ C 4 consist of the following covectors: e i , with multiplicity p, 1 ≤ i ≤ 3, e 4 , with multiplicity q, e i ± e j , with multiplicity r, 1 ≤ i < j ≤ 3, 1 2 (e 1 ± e 2 ± e 3 ± e 4 ), with multiplicity s, where p, q, r, s ∈ C are such that 4r + s = 0. Then A is a trigonometric ∨-system if p = 2r + s, and ps = 0. The corresponding prepotential (2.1) with is a solution of WDVV equations.
Let us now find the quadratic form G (7.5) By making further use of relations (7.1), (7.2) the expression (7.5) can be simplified to the form The final statement of the proposition follows from formulas (7.4), (7.6) and Theorem 2.9.
Remark 7.2. We note that for special values of the parameters configuration A is a restriction of a root system (cf. [17] where the rational version of this configuration was considered). Thus if r = 0 and p = q = s then A reduces to the root system D 4 . If r = 1, s = 4, then p = 6 and q = 1 and the resulting configuration is the restriction of the root system E 7 along subsystem of type A 3 . If s = 2r then the resulting configuration is the restriction of the root system E 6 along subsystem of type A 1 × A 1 .
Further solutions of WDVV equations can be obtained from Proposition 7.1 by restricting the configuration A. Proposition 7.3. Let A 1 ⊂ C 3 be the configuration with the corresponding multiplicities {r, 2p, p, q, 2r, 2s, s}, where p, q, r, s ∈ C. Let configuration A 2 ⊂ C 3 consist of the following set of covectors: e i + e j , with multiplicity r + s, 1 ≤ i < j ≤ 3, e i − e j , with multiplicity r, 1 ≤ i < j ≤ 3, e 1 + e 2 + e 3 , with multiplicity q + s.
Proof of this proposition follows from an observation that configuration A 1 can be obtained from the configuration A from Proposition 7.1 by restricting it to the hyperplane x 1 = x 2 (up to renaming the vectors). Similarly, configuration A 2 can be obtained by restricting the configuration A to the hyperplane x 1 + x 2 + x 3 − x 4 = 0 (and up to renaming the vectors). Other three-dimensional restrictions of the configuration A give restriction of the root system F 4 and a configuration from BC 3 family.
Rational versions of configurations A 1 , A 2 were considered in [17]. Note that configuration A 1 has collinear vectors 2e 1 , e 1 , so its rational version has different size.
Two-dimensional restrictions of A are considered below in Proposition 7.6 and Proposition 7.7, or can belong to BC 2 family of configuration, or have the form of configuration G 2 or appear in [19,Proposition 5].
Let us now consider examples of solutions (2.1) of WDVV equations where configuration A contains a small number of vectors on the plane. The next two propositions confirm that trigonometric ∨-systems with up to five covectors belong to A 2 or BC 2 families. Proposition 7.4. Any irreducible trigonometric ∨-system A ⊂ C 2 consisting of three vectors with non-zero multiplicities has the form (6.6) where n = 2 for some values of parameters.
Proof. By [19,Proposition 2] any such configuration has the form A = {α, β, γ} with the corresponding multiplicities {c α , c β , c γ }, where vectors in A satisfy α ± β ± γ = 0 for some choice of signs. It is easy to see that equations for m 1 , m 2 , m 3 , t ∈ C can be resolved.
Proposition 7.5. Any irreducible trigonometric ∨-system A ⊂ C 2 consisting of four or five vectors with non-zero multiplicities has the form BC 2 (r, s, q; m) for some values of parameters.
In order to compare the configuration B with the configuration BC 2 (r, s, q; m), we require parameters r, s, q, m 1 , m 2 to satisfy rm 1 = c 1 , rm 2 = c 2 , qm 1 m 2 = c + , These equations can be solved by taking In the rest of this section we give more examples of trigonometric ∨-systems on the plane, which can be checked directly or using Theorem 2.9. The configuration in the following proposition can be obtained by restricting configuration A 1 from Proposition 7.3 to the plane 2x 1 + x 2 − x 3 = 0.
The configuration in the following proposition can be obtained by restricting configuration A 1 from Proposition 7.3 to the plane x 3 = 0.

root systems solutions revisited
Following [21,25], recall that WDVV equations (2.2) have solutions of the form where R ⊂ V * is a root system of rank N, multiplicities c α and the inner product ·, · are invariant under the Weyl group, γ = γ (R,c) ∈ C and function f is given by (5.29). The corresponding values of γ (R,c) were given explicitly in [21,25] for constant multiplicity functions c α = t ∀α (except for R = BC N , G 2 ), they were found in [8] for special multiplicities and in [29,30] for arbitrary (non-reduced) root system R with invariant multiplicity. For type E root systems we have Similarly to analysis of the BC N case in Section 5 these solutions lead to solutions F of the form (2.1) for A = R + and the corresponding values of λ = λ (R,c) are given by We recall that λ (R,c) , in contrast to γ (R,c) , is invariant under linear transformations applied to R. An alternative way to derive values (8.2) is to apply Theorem 4.4 to already known solutions. Thus λ (E 6 ,t) can be derived, for example, by considering the four-dimensional restriction of E 6 along a subsystem of type A 1 × A 1 as this restriction is equivalent to the configuration from Proposition 7.1 when parameter s = 2r. Likewise restriction of E 7 along a subsystem of type A 3 gives the same configuration from Proposition 7.1 with r = 1 and s = 4. Similarly, restriction of E 8 along a subsystem of type D 6 gives the configuration of type BC 2 which allows to get λ (E 8 ,t) .
Let us now find λ (R,c) for the remaining cases, namely, R = F 4 and R = G 2 , and general multiplicity. We start with the root system R = F 4 . Proof. We note that the restriction of the configuration defined in Proposition 7.1 to the hyperplane x 4 = 0 gives the same configuration as one gets by restricting A =F + 4 to the hyperplane x 4 = 0. Hence λ is given by formula (7.3). Proposition 8.1 has the following implication for the corresponding solution of the form (8.1), which is also contained in [30]. c α f (α( x)), (8.5) where λ is given by (8.4), and we redenoted variables (x, y) by ( x, y). By dividing by −λ and changing variables x = −ix, y = γλ 6(s+2r) y, solution (8.5) takes the form (8.1) provided that γ 2 λ 2 = −108(s + 2r) 2 , which implies the statement.
Proposition 8.5. [8] The value of γ (R,c) in the solution (8.10) in the case of constant multiplicity function c α = t is given by Now we give a generalization of Proposition 8.5 to the case of non-constant multiplicity function. Let p be the multiplicity of short roots and q be the multiplicity of long roots in a reduced non-simply laced root system R. Proof. It follows from Proposition 5.4 that γ 2 (B N ,c) = −q(p + (N − 2)q). Note that θ B N = e 1 + e 2 = α 1 + 2(α 2 + · · · + α N ). Then it is easy to see that the substitution of (8.12) into formula (8.11) gives the same value of γ (B N ,c) . Similarly, we have γ 2 (C N ,c) = −p 2q + (N − 2)p ,